Additional File 3. Winbugs Code for Model M4: Covariate Adjusted NMA of IPD and Aggregate Data

Additional file 3. WinBUGS code for Model M4: Covariate adjusted NMA of IPD and aggregate data

Our WinBUGS [56] code for the model M3 described in Section 2.3, with comments and selected inputs, is provided below. Omitting the covariates and or only using AD will obtain models M1 and M2. Covariates are omitted or included by setting across.cov[j], within.cov[j], or inter.cov[j] to 0 or 1, respectively, where j numbers the possible covariates, six in this example. Appropriately setting Naggregate or Nipd, the number of aggregate or ipd studies, respectively, will set the analysis to AD, IPD or a combination of both.

As the IPD was proprietary, only the AD of the studies provided below is real while the IPD is simulated. For the IPD, we generated four two-arm trials with five patients in each arm and investigating the treatment combinations of the original IMPRES study. The covariate values for age, sex, baseline walk distance, baseline MPAP, baseline NYHA STATUS and baseline PVR were generated from distributions similar to those of the original IPD. However, re-running the analyses will not generate similar results to those presented in the Table 6 as our simulated data assumed no relation between treatments and outcomes and is merely illustrative. Several tricks were employed to implement the model and are explained below. We are happy to answer any queries about our code.

The WinBUGS language does not have the ‘if’ and ‘else’ construction commonly found in other programming languages. We overcame this absence through the use of intermediary and indicator variables. For example, if study i was observational, indicated by observational[i]=1, we did not want to include its control arm data YC[i] (the short term change in 6MWD). We would like to able to write (in WinBUGS pseudo-code)

if(!observational[j])YC[j]~dnorm(muC[j],tauC[j]

so that the model parameter muC[j] was only connected to the data YC[j] if the study was not observational. We accomplish the same functionality through an intermediary variable muCtemp[j] which is always connected to the data but is only equal to muC[j] if the study is not observational, i.e.

YC[j]~dnorm(muCtemp[j],tauC[j])

muCtemp[j]<-(1-observational[j])*muC[j]

This method is also used for conditional inclusion of covariate effects. For example, within-study covariate effects pi[j], for covariate j, were included only if the indicator within.cov[j] was 1 and this was linked to pi[j] through the intermediary variable pi.temp[j].

One further note is that WinBUGS cannot manage very long program lines, so we broke our equations across several lines, for example in the covariate effect equations.

# Network Meta regression model including IPD and aggregate data studies

# Includes covariate effects and interaction terms

model

{

# Aggregate data from literature review

for (j in 1:Naggregate) { # Naggregate is number of included aggregate data studies

# delta are the quality weights (delta=1 means no internal bias, delta=0.1 is our down-weighting factor for model S1)

tauA[j] <- deltaA[j]/(seA[j] * seA[j])# precision in alternate arm

tauC[j] <- deltaC[j]/(seC[j] * seC[j])# precision in control arm

YA[j] ~ dnorm(muA[j], tauA[j])# Change from baseline in alternate arm

YC[j] ~ dnorm(muCtemp[j], tauC[j])# Change from baseline in control arm

muCtemp[j] <- (1 - observational[j]) * muC[j] # If the study is observational, muC[j] is not connected to the (missing) data

muA[j] <- alpha[j] + phi[1] * mean.age[2, j] + phi[2] * mean.sex[2, j] + phi[3] * mean.status[2, j] + phi[4] * mean.walk[2, j] + phi[5] * mean.pvr[2, j] + theta[1, j] + phi[6] * mean.mpap[2, j]

muC[j] <- alpha[j] + phi[1] * mean.age[1, j] + phi[2] * mean.sex[1, j] + phi[3] * mean.status[1, j] + phi[4] * mean.walk[1, j] + phi[5] * mean.pvr[1, j] + theta.control[1, j] + phi[6] * mean.mpap[1, j]

}

# Prior is on the standard deviation scale

tau <- 1/(sd * sd)

sd ~ dunif(0.00000E+00, upper.lsd)

# IPD data in paper is from IMPRES study but here is only simulated

for (j in 1:Nipd) {# Nipd is number of included IPD studies

for (i in 1:N[j]) {# N[j] is number of patients in study j

Y[i, j] ~ dnorm(mu[i, j], tau)# Change from baseline 6MWD for patient i

# It was necessary to split the (long) model equation for WinBUGS to work

mu[i, j] <- alpha[j + Naggregate] + theta[i, j + Naggregate] * (treat[i, j] - 1) + theta.control[i, j + Naggregate] * treat[i, j] + covariates.across[i, j] + covariates.within[i, j]

# Across-study covariate effects on the main effect

covariates.across[i, j] <- phi[1] * mean.age[treat[i, j], j + Naggregate] + phi[2] * mean.sex[treat[i, j], j + Naggregate] + phi[3] * mean.status[treat[i, j], j + Naggregate] + phi[4] * mean.walk[treat[i, j], j + Naggregate] + phi[5] * mean.pvr[treat[i, j], j + Naggregate] + phi[6] * mean.mpap[treat[i, j], j + Naggregate]

# Within-study covariate effects on the main effect

covariates.within[i, j] <- pi[1] * (age[i, j] - mean.age[treat[i, j], j + Naggregate]) + pi[2] * (sex[i, j] - mean.sex[treat[i, j], j + Naggregate]) + pi[3] * (status[i, j] -mean.status[treat[i, j], j + Naggregate]) + pi[4] * (walk[i, j] - mean.walk[treat[i, j], j + Naggregate]) + pi[5] * (pvr[i, j] - mean.pvr[treat[i, j], j + Naggregate]) + pi[6] * (mpap[i, j] - mean.mpap[treat[i, j], j + Naggregate])

}

# Effect of additional treatments in aggregate studies

# beta[i] are the treatment effects

# study.beta[i,j] indicates whether treatment i was used as add-on therapy in study j

for (j in 1:Naggregate) {

theta.mean.agg[1, j] <- beta[1] * study.beta[1, j]

theta.control.mean.agg[1, j] <- beta[1] * study.beta.control[1, j]

for (i in 2:N.beta) {

theta.mean.agg[i, j] <- theta.mean.agg[i - 1, j] + beta[i] * study.beta[i, j]

theta.control.mean.agg[i, j] <- theta.control.mean.agg[i - 1, j] + beta[i] * study.beta.control[i, j]

}

theta[1, j] ~ dnorm(theta.mean.agg[N.beta, j], beta.tau)

theta.control[1, j] ~ dnorm(theta.control.mean.agg[N.beta, j], beta.tau)

}

for (j in (Naggregate + 1):(Nipd + Naggregate)) {

for (i in 1:N[j - Naggregate]) {

theta[i, j] ~ dnorm(theta.mean.ipd[N.beta, i, j], beta.tau)

theta.control[i, j] ~ dnorm(theta.control.mean.ipd[N.beta, i, j], beta.tau)

# Place all within-study treatment covariate interactions here

theta.mean.ipd[1, i, j] <- study.beta[1, j] * (beta[1] + gamma[1, 1] * (age[i, j - Naggregate] - mean.age[treat[i, j - Naggregate], j]) + gamma[1, 2] * (sex[i, j - Naggregate] - mean.sex[treat[i, j - Naggregate], j]) + gamma[1, 3] * (status[i, j - Naggregate] - mean.status[treat[i, j - Naggregate], j]) + gamma[1, 4] * (walk[i, j - Naggregate] - mean.walk[treat[i, j - Naggregate], j]) + gamma[1, 5] * (pvr[i, j - Naggregate] - mean.pvr[treat[i, j - Naggregate], j]) + gamma[1, 6] * (mpap[i, j - Naggregate] - mean.mpap[treat[i, j - Naggregate], j]))

theta.control.mean.ipd[1, i, j] <- study.beta.control[1, j] * (beta[1] + gamma[1, 1] * (age[i, j - Naggregate] - mean.age[treat[i, j - Naggregate], j]) + gamma[1, 2] * (sex[i, j - Naggregate] - mean.sex[treat[i, j - Naggregate], j]) + gamma[1, 3] * (status[i, j - Naggregate] - mean.status[treat[i, j - Naggregate], j]) + gamma[1, 4] * (walk[i, j - Naggregate] - mean.walk[treat[i, j - Naggregate], j]) + gamma[1, 5] * (pvr[i, j - Naggregate] - mean.pvr[treat[i, j - Naggregate], j]) + gamma[1, 6] * (mpap[i, j - Naggregate] - mean.mpap[treat[i, j - Naggregate], j]))

for (k in 2:N.beta) {

theta.mean.ipd[k, i, j] <- theta.mean.ipd[k - 1, i, j] + study.beta[k, j] * (beta[k] + gamma[k, 1] * (age[i, j - Naggregate] - mean.age[treat[i, j - Naggregate], j]) + gamma[k, 2] * (sex[i, j - Naggregate] - mean.sex[treat[i, j - Naggregate], j]) + gamma[k, 3] * (status[i, j - Naggregate] - mean.status[treat[i, j - Naggregate], j]) + gamma[k, 4] * (walk[i, j - Naggregate] - mean.walk[treat[i, j - Naggregate], j]) + gamma[k, 5] * (pvr[i, j - Naggregate] - mean.pvr[treat[i, j - Naggregate], j]) + gamma[k, 6] * (mpap[i, j - Naggregate] - mean.mpap[treat[i, j - Naggregate], j]))

theta.control.mean.ipd[k, i, j] <- theta.control.mean.ipd[k - 1, i, j] + study.beta.control[k, j] * (beta[k] + gamma[k, 1] * (age[i, j - Naggregate] - mean.age[treat[i, j - Naggregate], j]) + gamma[k, 2] * (sex[i, j - Naggregate] - mean.sex[treat[i, j - Naggregate], j]) + gamma[k, 3] * (status[i, j - Naggregate] - mean.status[treat[i, j - Naggregate], j]) + gamma[k, 4] * (walk[i, j - Naggregate] - mean.walk[treat[i, j - Naggregate], j]) + gamma[k, 5] * (pvr[i, j - Naggregate] - mean.pvr[treat[i, j - Naggregate], j]) + gamma[k, 6] * (mpap[i, j - Naggregate] - mean.mpap[treat[i, j - Naggregate], j]))

}

# Random effects on the baseline change in 6MWD (random.effects=1 if using a random effect, otherwise uses fixed effect alpha.mean)

for (j in 1:(Nipd + Naggregate)) {

alpha.random.effect[j] ~ dnorm(alpha.mean, alpha.tau)

alpha[j] <- (1 - random.effects) * alpha.mean + random.effects *

alpha.random.effect[j]

}

# Prior on baseline change in 6MWD

alpha.mean ~ dnorm(0.00000E+00, mean.vague.prec)

# Place a uniform prior on the sd of alpha, as advised by Lambert et al

alpha.tau <- 1/(alpha.sd * alpha.sd)

alpha.sd ~ dunif(0.00000E+00, upper.lsd)

# Priors for the treatment effects (with random effects)

for (j in 1:N.beta) {

beta[j] ~ dnorm(0.00000E+00, vague.prec)

}

# Use a uniform prior on the standard deviation

beta.tau <- 1/(beta.sd * beta.sd)

beta.sd ~ dunif(0.00000E+00, upper.lsd)

# Priors for all 6possible covariate effects

for (j in 1:6) {

pi.temp[j] ~ dnorm(0.00000E+00, vague.prec)# Within study effect

pi[j] <- within.cov[j] * pi.temp[j]#pi[j] is 0 if effect is not included at within-study level

phi.temp[j] ~ dnorm(0.00000E+00, mean.vague.prec)# Effect of study mean

phi[j] <- across.cov[j] * phi.temp[j]# phi[j] is 0 if effect is not included at across-study level

}

# Priors for the 6 possible treatment covariate interaction effects at within-study level

for (j in 1:N.beta) {

for (i in 1:6) {

gamma.temp[j, i] ~ dnorm(0.00000E+00, vague.prec)

gamma[j, i] <- inter.cov[i] * gamma.temp[j, i]# gamma[j,i] is 0 if effect not included

}

##############################################################################################################

## Two sets of initial values for placebo effects ###############################

list(alpha=c(0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00))

##################################################################################################################

## Illustrative data – AD is real while IPD is simulated ##############################

list(Naggregate=1.50000E+01, Nipd=4.00000E+00, Y= structure(.Data= c(-9.00000E+01, 1.00000E+00, 2.30000E+01, 5.90000E+01, 5.00000E+01, 6.50000E+01, 1.04000E+02, 1.04000E+02, 6.00000E+01, 1.04000E+02, -6.00000E+00, -5.00000E+01, -3.40000E+01, 1.00000E+01, 3.00000E+01, -2.40000E+01, 6.00000E+00, 4.70000E+01, 8.40000E+01, 3.20000E+01, -1.20000E+01, 1.20000E+01, -4.50000E+01, -5.40000E+01, -2.40000E+01, -2.20000E+01, 9.10000E+01, 6.60000E+01, 4.20000E+01, -8.70000E+01, 6.00000E+00, 9.10000E+01, 1.00000E+01, -1.40000E+01, 3.80000E+01, -5.20000E+01, -5.80000E+01, 8.00000E+01, 6.60000E+01, -6.60000E+01), .Dim=c(10, 4)), treat= structure(.Data= c(1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 1.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00, 2.00000E+00), .Dim=c(10, 4)), N=c(1.00000E+01, 1.00000E+01, 1.00000E+01, 1.00000E+01), age= structure(.Data= c(6.20000E+01, 7.20000E+01, 3.30000E+01, 5.30000E+01, 5.80000E+01, 3.40000E+01, 8.20000E+01, 5.20000E+01, 4.20000E+01, 4.90000E+01, 3.90000E+01, 3.90000E+01, 5.10000E+01, 5.60000E+01, 7.50000E+01, 3.00000E+01, 4.10000E+01, 4.40000E+01, 4.90000E+01, 3.50000E+01, 7.20000E+01, 2.90000E+01, 4.20000E+01, 4.50000E+01, 2.10000E+01, 3.80000E+01, 4.80000E+01, 3.50000E+01, 8.50000E+01, 6.00000E+01, 5.80000E+01, 5.80000E+01, 6.40000E+01, 4.00000E+01, 4.80000E+01, 2.00000E+01, 5.50000E+01, 2.80000E+01, 6.60000E+01, 5.10000E+01), .Dim=c(10, 4)), sex= structure(.Data= c(1.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 1.00000E+00, 0.00000E+00, 0.00000E+00, 1.00000E+00, 0.00000E+00, 0.00000E+00, 1.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 1.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 1.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00, 0.00000E+00), .Dim=c(10, 4)), mpap= structure(.Data= c(4.70000E+01, 6.00000E+01, 4.80000E+01, 5.60000E+01, 8.70000E+01, 7.10000E+01, 6.50000E+01, 3.10000E+01, 5.80000E+01, 4.80000E+01, 6.50000E+01, 7.30000E+01, 5.70000E+01, 4.90000E+01, 5.30000E+01, 4.50000E+01, 7.30000E+01, 5.80000E+01, 7.20000E+01, 4.10000E+01, 7.40000E+01, 3.70000E+01, 4.70000E+01, 6.00000E+01, 4.70000E+01, 7.10000E+01, 6.10000E+01, 4.90000E+01, 5.10000E+01, 5.20000E+01, 5.10000E+01, 6.30000E+01, 6.60000E+01, 3.90000E+01, 5.50000E+01, 2.80000E+01, 9.20000E+01, 5.80000E+01, 6.20000E+01, 5.30000E+01), .Dim=c(10, 4)), status= structure(.Data= c(2.00000E+00, 2.00000E+00, 2.00000E+00, 3.00000E+00, 2.00000E+00, 3.00000E+00, 3.00000E+00, 2.00000E+00, 3.00000E+00, 4.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 4.00000E+00, 3.00000E+00, 2.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 2.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 3.00000E+00, 2.00000E+00, 3.00000E+00, 4.00000E+00, 3.00000E+00, 2.00000E+00, 2.00000E+00, 3.00000E+00, 3.00000E+00, 4.00000E+00, 3.00000E+00, 3.00000E+00, 4.00000E+00), .Dim=c(10, 4)), walk= structure(.Data= c(4.14000E+02, 2.90000E+02, 3.95000E+02, 3.78000E+02, 3.29000E+02, 4.23000E+02, 2.46000E+02, 3.30000E+02, 2.36000E+02, 2.47000E+02, 5.56000E+02, 3.45000E+02, 4.51000E+02, 4.47000E+02, 3.79000E+02, 4.25000E+02, 2.68000E+02, 2.76000E+02, 3.84000E+02, 2.26000E+02, 4.83000E+02, 3.03000E+02, 2.89000E+02, 3.43000E+02, 4.20000E+02, 4.66000E+02, 3.30000E+02, 3.33000E+02, 3.28000E+02, 3.83000E+02, 3.18000E+02, 3.80000E+02, 2.27000E+02, 3.49000E+02, 3.18000E+02, 3.63000E+02, 5.04000E+02, 2.55000E+02, 3.33000E+02, 4.54000E+02), .Dim=c(10, 4)), pvr= structure(.Data= c(1.19700E+03, 9.51000E+02, 1.63000E+03, 5.55000E+02, 1.56500E+03, 1.01600E+03, 4.15000E+02, 1.41000E+03, 1.10200E+03, 1.59800E+03, 8.79000E+02, 1.26600E+03, 1.23800E+03, 1.47900E+03, 1.31000E+03, 5.10000E+02, 1.08600E+03, 2.98000E+02, 1.05000E+03, 6.44000E+02, 1.23700E+03, 1.36400E+03, 1.34900E+03, 1.26300E+03, 1.75700E+03, 1.23500E+03, 9.40000E+01, 1.28800E+03, 4.52000E+02, 6.94000E+02, 6.45000E+02, 7.52000E+02, 1.12500E+03, 1.83400E+03, 5.19000E+02, 1.14700E+03, 1.16300E+03, 1.19100E+03, 1.23800E+03, 1.61500E+03), .Dim=c(10, 4)), YA=c(4.10000E+01, 3.00000E+00, 6.70000E+01, 5.80000E+01, 1.96000E+01, 5.70000E+01, 4.02000E+01, 2.98000E+01, 3.00000E+01, 7.20000E+01, 7.00000E+01, 3.60000E+01, 2.30000E+01, 3.20000E+01, 5.00000E+01, -8.40000E+00, -6.20000E+00, 3.12000E+01, -3.00000E+00), YC=c(-2.50000E+01, -2.50000E+01, -2.50000E+01, -2.50000E+01, -2.50000E+01, -2.50000E+01, 1.88000E+01, 1.00000E+00, 4.00000E+00, 4.60000E+01, -6.00000E+00, -8.00000E+00, -6.50000E+00, -1.50000E+01, 2.00000E+00, -1.60000E+00, 4.54000E+01, 4.70000E+01, 2.42000E+01), seA=c(3.78021E+01, 2.29824E+01, 4.47590E+01, 9.61509E+00, 2.84582E+01, 3.45704E+01, 8.50000E+00, 5.31322E+00, 1.02899E+01, 1.14708E+01, 2.34135E+01, 6.54535E+00, 9.32059E+00, 2.47588E+01, 9.00000E+00, 1.67260E+01, 2.70193E+01, 2.37432E+01, 3.35917E+01), seC=c(3.78021E+01, 2.29824E+01, 4.47590E+01, 9.61509E+00, 2.84582E+01, 3.45704E+01, 9.15000E+00, 5.29336E+00, 1.06187E+01, 1.96061E+01, 5.04705E+01, 9.45560E+00, 9.16903E+00, 3.32415E+01, 5.00000E+00, 2.77301E+01, 1.87686E+01, 2.03617E+01, 2.78108E+01), mean.age= structure(.Data= c(3.70000E+01, 3.20000E+01, 5.12000E+01, 4.60000E+01, 5.61600E+01, 3.90000E+01, 5.17000E+01, 4.75000E+01, 5.10000E+01, 4.70000E+01, 4.74000E+01, 4.72000E+01, 5.30000E+01, 4.00000E+01, 4.90000E+01, 5.08000E+01, 5.10000E+01, 5.56000E+01, 4.18000E+01, 3.70000E+01, 3.20000E+01, 5.12000E+01, 4.60000E+01, 5.61600E+01, 3.90000E+01, 5.00000E+01, 4.78000E+01, 4.90000E+01, 4.50000E+01, 5.22000E+01, 4.87000E+01, 4.90000E+01, 4.00000E+01, 4.80000E+01, 5.94000E+01, 3.90000E+01, 5.24000E+01, 4.18000E+01), .Dim=c(2, 19)), mean.sex= structure(.Data= c(1.87500E-01, 1.25000E-01, 9.00000E-02, 3.00000E-01, 4.00000E-02, 2.22222E-01, 2.20000E-01, 2.30000E-01, 2.10000E-01, 4.50000E-01, 0.00000E+00, 2.20000E-01, 2.40000E-01, 3.00000E-01, 1.90000E-01, 2.00000E-01, 2.00000E-01, 4.00000E-01, 2.00000E-01, 1.87500E-01, 1.25000E-01, 9.00000E-02, 3.00000E-01, 4.00000E-02, 2.22222E-01, 2.10000E-01, 1.80000E-01, 2.10000E-01, 2.30000E-01, 1.90000E-01, 2.10000E-01, 2.20000E-01, 2.40000E-01, 2.10000E-01, 0.00000E+00, 0.00000E+00, 2.00000E-01, 0.00000E+00), .Dim=c(2, 19)), mean.mpap= structure(.Data= c(5.60000E+01, 8.01000E+01, 4.90000E+01, 5.50000E+01, 5.50800E+01, 6.20000E+01, 5.14534E+01, 5.11000E+01, 5.20000E+01, 6.09000E+01, 5.60000E+01, 5.30000E+01, 4.90000E+01, 5.90000E+01, 5.60000E+01, 6.44000E+01, 5.72000E+01, 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